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                <h1 id="regularization">Regularization</h1>
<p>In their book <a href="#deep-learning">Deep Learning</a> Ian Goodfellow et al. define regularization as</p>
<blockquote>
<p>"any modification we make to a learning algorithm that is intended to reduce its generalization error, but not its training error."</p>
</blockquote>
<p>PyTorch's <a href="http://pytorch.org/docs/master/optim.html">optimizers</a> use \(l_2\) parameter regularization to limit the capacity of models (i.e. reduce the variance).</p>
<p>In general, we can write this as:
\[
loss(W;x;y) = loss_D(W;x;y) + \lambda_R R(W)
\]
And specifically,
\[
loss(W;x;y) = loss_D(W;x;y) + \lambda_R \lVert W \rVert_2^2
\]
Where W is the collection of all weight elements in the network (i.e. this is model.parameters()), \(loss(W;x;y)\) is the total training loss, and \(loss_D(W)\) is the data loss (i.e. the error of the objective function, also called the loss function, or <code>criterion</code> in the Distiller sample image classifier compression application).</p>
<pre><code>optimizer = optim.SGD(model.parameters(), lr = 0.01, momentum=0.9, weight_decay=0.0001)
criterion = nn.CrossEntropyLoss()
...
for input, target in dataset:
    optimizer.zero_grad()
    output = model(input)
    loss = criterion(output, target)
    loss.backward()
    optimizer.step()
</code></pre>

<p>\(\lambda_R\) is a scalar called the <em>regularization strength</em>, and it balances the data error and the regularization error.  In PyTorch, this is the <code>weight_decay</code> argument.</p>
<p>\(\lVert W \rVert_2^2\) is the square of the \(l_2\)-norm of W, and as such it is a <em>magnitude</em>, or sizing, of the weights tensor.
\[
\lVert W \rVert_2^2 = \sum_{l=1}^{L}  \sum_{i=1}^{n} |w_{l,i}|^2 \;\;where \;n = torch.numel(w_l)
\]</p>
<p>\(L\) is the number of layers in the network; and the notation about used 1-based numbering to simplify the notation.</p>
<p>The qualitative differences between the \(l_2\)-norm, and the squared \(l_2\)-norm is explained in <a href="https://www.deeplearningbook.org/">Deep Learning</a>.</p>
<h2 id="sparsity-and-regularization">Sparsity and Regularization</h2>
<p>We mention regularization because there is an interesting interaction between regularization and some DNN sparsity-inducing methods.</p>
<p>In <a href="#han-et-al-2017">Dense-Sparse-Dense (DSD)</a>, Song Han et al. use pruning as a regularizer to improve a model's accuracy:</p>
<blockquote>
<p>"Sparsity is a powerful form of regularization. Our intuition is that, once the network arrives at a local minimum given the sparsity constraint, relaxing the constraint gives the network more freedom to escape the saddle point and arrive at a higher-accuracy local minimum."</p>
</blockquote>
<p>Regularization can also be used to induce sparsity.  To induce element-wise sparsity we can use the \(l_1\)-norm, \(\lVert W \rVert_1\).
\[
\lVert W \rVert_1 = l_1(W) = \sum_{i=1}^{|W|} |w_i|
\]</p>
<p>\(l_2\)-norm regularization reduces overfitting and improves a model's accuracy by shrinking large parameters, but it does not force these parameters to absolute zero.  \(l_1\)-norm regularization sets some of the parameter elements to zero, therefore limiting the model's capacity while making the model simpler.  This is sometimes referred to as <em>feature selection</em> and gives us another interpretation of pruning.</p>
<p><a href="https://github.com/NervanaSystems/distiller/blob/master/jupyter/L1-regularization.ipynb">One</a> of Distiller's Jupyter notebooks explains how the \(l_1\)-norm regularizer induces sparsity, and how it interacts with \(l_2\)-norm regularization.</p>
<p>If we configure <code>weight_decay</code> to zero and use \(l_1\)-norm regularization, then we have:
\[
loss(W;x;y) = loss_D(W;x;y) + \lambda_R \lVert W \rVert_1
\]
If we use both regularizers, we have:
\[
loss(W;x;y) = loss_D(W;x;y) + \lambda_{R_2} \lVert W \rVert_2^2  + \lambda_{R_1} \lVert W \rVert_1
\]</p>
<p>Class <code>distiller.L1Regularizer</code> implements \(l_1\)-norm regularization, and of course, you can also schedule regularization.</p>
<pre><code>l1_regularizer = distiller.s(model.parameters())
...
loss = criterion(output, target) + lambda * l1_regularizer()
</code></pre>

<h2 id="group-regularization">Group Regularization</h2>
<p>In Group Regularization, we penalize entire groups of parameter elements, instead of individual elements.  Therefore, entire groups are either sparsified (i.e. all of the group elements have a value of zero) or not.  The group structures have to be pre-defined.</p>
<p>To the data loss, and the element-wise regularization (if any), we can add group-wise regularization penalty.  We represent all of the parameter groups in layer \(l\) as \( W_l^{(G)} \), and we add the penalty of all groups for all layers.  It gets a bit messy, but not overly complicated:
\[
loss(W;x;y) = loss_D(W;x;y) + \lambda_R R(W) + \lambda_g \sum_{l=1}^{L} R_g(W_l^{(G)})
\]</p>
<p>Let's denote all of the weight elements in group \(g\) as \(w^{(g)}\).</p>
<p>\[
R_g(w^{(g)}) = \sum_{g=1}^{G} \lVert w^{(g)} \rVert_g = \sum_{g=1}^{G} \sum_{i=1}^{|w^{(g)}|} {(w_i^{(g)})}^2
\]
where \(w^{(g)} \in w^{(l)} \) and \( |w^{(g)}| \) is the number of elements in \( w^{(g)} \).</p>
<p>\( \lambda_g \sum_{l=1}^{L} R_g(W_l^{(G)}) \) is called the Group Lasso regularizer.  Much as in \(l_1\)-norm regularization we sum the magnitudes of all tensor elements, in Group Lasso we sum the magnitudes of element structures (i.e. groups).<br />
<br>
Group Regularization is also called Block Regularization, Structured Regularization, or coarse-grained sparsity (remember that element-wise sparsity is sometimes referred to as fine-grained sparsity).  Group sparsity exhibits regularity (i.e. its shape is regular), and therefore
it can be beneficial to improve inference speed.</p>
<p><a href="#huizi-et-al-2017">Huizi-et-al-2017</a> provides an overview of some of the different groups: kernel, channel, filter, layers.  Fiber structures such as matrix columns and rows, as well as various shaped structures (block sparsity), and even <a href="#anwar-et-al-2015">intra kernel strided sparsity</a> can also be used.</p>
<p><code>distiller.GroupLassoRegularizer</code> currently implements most of these groups, and you can easily add new groups.</p>
<h2 id="references">References</h2>
<p><div id="deep-learning"></div> <strong>Ian Goodfellow and Yoshua Bengio and Aaron Courville</strong>.
    <a href="https://www.deeplearningbook.org/"><em>Deep Learning</em></a>,
     arXiv:1607.04381v2,
    2017.</p>
<div id="han-et-al-2017"></div>

<p><strong>Song Han, Jeff Pool, Sharan Narang, Huizi Mao, Enhao Gong, Shijian Tang, Erich Elsen, Peter Vajda, Manohar Paluri, John Tran, Bryan Catanzaro, William J. Dally</strong>.
    <a href="https://arxiv.org/abs/1607.04381"><em>DSD: Dense-Sparse-Dense Training for Deep Neural Networks</em></a>,
     arXiv:1607.04381v2,
    2017.</p>
<div id="huizi-et-al-2017"></div>

<p><strong>Huizi Mao, Song Han, Jeff Pool, Wenshuo Li, Xingyu Liu, Yu Wang, William J. Dally</strong>.
    <a href="https://arxiv.org/abs/1705.08922"><em>Exploring the Regularity of Sparse Structure in Convolutional Neural Networks</em></a>,
    arXiv:1705.08922v3,
    2017.</p>
<div id="anwar-et-al-2015"></div>

<p><strong>Sajid Anwar, Kyuyeon Hwang, and Wonyong Sung</strong>.
    <a href="https://arxiv.org/abs/1512.08571"><em>Structured pruning of deep convolutional neural networks</em></a>,
    arXiv:1512.08571,
    2015</p>
              
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